It’s been more than 60 years since Edmund Hillary (later Sir Edmund, of course) and Tenzing Norgay became the first climbers to summit Mount Everest, and all these decades later their feat still resonates as one of the 20th century’s signature moments. Here, LIFE.com looks back at that remarkable time with some rare photos from the celebrations after the climb, as well as page spreads from the cover story that ran in LIFE a few months later chronicling the accomplishment — and the bitter controversy that swirled around the entire event.

As LIFE noted in its July 13, 1953, issue, the historic ascent was hardly greeted with unalloyed goodwill and enthusiasm from all corners of the globe. In fact, international politics and racial pride were quickly thrust into the conversation about Hillary’s and Tenzing’s astonishing feat.

“Everest’s Conqueror’s Come Back,” LIFE roared in one headline in that special issue, then immediately blunted the celebratory tone with a caveat: “They bring thrilling stories of a great deed, but little men besmirch their riotous welcome.”

Thus — in a sad foreshadowing of the often contentious debate that had dogged so many attempts on Everest throughout the years (Is it worth the risk of life and limb? What does the local community get out of it?) — the very first successful climb to the top of the world’s highest peak sparked some often quite ugly jockeying for credit and supremacy. Jockeying, it should be noted, that both Hillary and Tenzing, who were fast friends, readily denounced.

(Also, while LIFE makes more than one mention of “British climbers” in its reporting, Edmund Hillary was in fact a proud, born-and-raised New Zealander. He died in 2008, at the age of 88, in Auckland. Tenzing died two years before Hillary, at age 71, in India.)

“The climbers who conquered Everest,” LIFE wrote, “came down to a world eager to see them, honor them and hear their full story. . . . They came down to such a welcome — such surging excitement and hero worship — as had never before stirred the steamy lowlands of Nepal.”

The first official welcomers met the mountaineers outside of Banepa [the article continues], 20 miles from Nepal’s capital Katmandu. In the lead was British embassy party, bearing beer and sandwiches; then came the Nepalese to garland the heroes with flowers and sprinkle them with kumkum, a vermilion powder of rejoicing. Devil dancers met that at Bhadgaon, still 15 miles out. The wife of Sir John Hunt, the expedition’s leader, came out to meet him. Tenzing’s wife and their two teenage daughters flew from Darjeeling, India. . . .

To the distress and the half-resentful bewilderment of Colonel Hunt and his British climbers, however, these first wild welcomings carried a clear implication that, in Asia, the real hero of Everest was Tenzing alone. The conquest of Everest, a product of selfless teamwork between Asian and European, was being twisted into an ugly tool of Asian nationalism, inflamed further by the normal British habit of treating the hired Tenzing like a hired man. . . .

Today, as men and women continue to test their own mettle on the peaks of the Himalayas and on the heights of other, equally lethal mountain ranges around the globe — occasionally losing toes, noses, fingers and even lives in the process — the pictures in this gallery are a reminder that for some people, the risks have always, unquestionably, been worth it.